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Theorem 0ss 3320
Description: The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
0ss  |-  (/)  C_  A

Proof of Theorem 0ss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 noel 3290 . . 3  |-  -.  x  e.  (/)
21pm2.21i 610 . 2  |-  ( x  e.  (/)  ->  x  e.  A )
32ssriv 3029 1  |-  (/)  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1438    C_ wss 2999   (/)c0 3286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287
This theorem is referenced by:  ss0b  3321  ssdifeq0  3363  sssnr  3595  ssprr  3598  uni0  3678  int0el  3716  0disj  3840  disjx0  3842  tr0  3945  0elpw  3997  fr0  4176  elnn  4418  rel0  4558  0ima  4787  fun0  5066  f0  5195  oaword1  6224  0domg  6543  nnnninf  6796  exmidfodomrlemim  6817  sum0  10767  0opn  11486  bdeq0  11641  bj-omtrans  11734  el2oss1o  11770  nninfsellemsuc  11787
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